I think a good entry point for making public healthcare better would be making each doctor visit require a small amount of money. E.g. $1 or $5. This way people that go to doctor just to have a talk (because they are bored - yes that happens in public healthcare, because it is free so people abuse it) would free up the queue for people that really need the visit.

AFAIR there was such proposition in France some time ago, I don't know how they solved that.